Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay

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Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices

and exponential decay

Fazia Bedouhene, Islam Boussaada, Silviu-Iulian Niculescu

To cite this version:

Fazia Bedouhene, Islam Boussaada, Silviu-Iulian Niculescu. Real spectral values coexistence and their

effect on the stability of time-delay systems: Vandermonde matrices and exponential decay. Comptes

Rendus. Mathématique, Centre Mersenne (2020-..) ; Elsevier Masson (2002-2019), 2020, 358 (9-10),

pp.1011-1032. �10.5802/crmath.112�. �hal-02476403�

Real spectral values coexistence and their effect on the stability of time-delay systems:

Vandermonde matrices and exponential decay

Fazia Bedouhene

Laboratoire de Math´ematiques Pures et Appliqu´ees, Mouloud Mammeri University of Tizi-Ouzou, Algeria

Islam Boussaada

Inria Saclay-ˆIle-de-France, Equipe DISCO & IPSA Ivry sur Seine & L2S, CNRS-CentraleSup´elec-Universit´e Paris Sud, Universit´e Paris Saclay, 91192 Gif-sur-Yvette cedex, France

Silviu-Iulian Niculescu

CNRS-CentraleSup´elec-Universit´e Paris Sud, Universit´e Paris Saclay, 91192 Gif-sur-Yvette cedex, France

Abstract

This work exploits structural properties of a class of functional Vandermonde matrices to emphasize some qual- itative properties of a class of linear autonomousn−th order differential equation with forcing term consisting in the delayed dependent-variable. More precisely, it deals with the stabilizing effect of delay parameter cou- pled with the coexistence of the maximal number of real spectral values. The derived conditions are necessary and sufficient and represent a novelty in the litterature. Under appropriate conditions, such a configuration characterizes the spectral abscissa corresponding to the studied equation. A new stability criterion is proposed.

This criterion extends recent results in factorizing quasipolynomial functions. The applicative potential of the proposed method is illustrated through the stabilization of coupled oscillators.

Keywords: Time-delay, asymptotic stability, exponential stability, exponential decay rate, Vandermonde matrix, quasipolynomial factorization, control design.

2010 MSC: 34K20, 39B82, 70Q05, 47N70

Contents

1 Introduction 2

E-mail addresses: fazia.bedouhene@ummto.dz (Fazia Bedouhene)., Islam.Boussaada@l2s.centralesupelec.fr ( Islam Boussaada)., Silviu.Niculescu@l2s.centralesupelec.fr (Silviu-Iulian Niculescu).

2 Problem settings and prerequisites 4

2.1 Counting quasipolynomial roots in horizontal strips . . . 5

2.2 Structured matrices appearing in the control of dynamical systems . . . 6

2.3 The determinant of a structured functional Vandermonde type matrix . . . 7

2.4 Symmetry property . . . 8

2.5 Shifting properties . . . 8

2.6 Factorization property . . . 9

3 Main results 9 3.1 Coexistence of n+ 1 real roots of ∆_n . . . 9

3.2 On qualitative properties ofs₁ as a root of ∆_n . . . 11

3.3 Exponential stability . . . 13 4 Stabilizing coupled oscillators using delayed output feedback 13

5 Concluding remarks 14

1. Introduction

Matrices arising from a wide range of problems in mathematics and engineering typically display characteris- tic structures. In particular, exploiting such a structure in problems from dynamical systems is known to be an engaging aperture for understanding of complex qualitative behaviors and for characterizing system’s properties, see, for instance,¹and references therein. This study is a crossroad between the investigation of the invertibility of a class of such structured matrices which is related toMultivariate Interpolation Problems (namely, the well- knownLagrange Interpolation Problem) and the localisation ofspectral valuesof lineartime-delay systems. The study of conditions on the time-delay systems parameters that guarantees the exponential stability of solutions is a question of ongoing interest and to the best of the authors’ knowledge it remains an open problem. In particular, in frequency-domain, the problem reduces to the analysis of the distribution of the roots of the corresponding characteristic equation, which is an entire function calledcharacteristic quasipolynomial), see for instance2,3,4,5,6,7,8.

The starting point of the present work is a property, discussed in recent studies, calledMultiplicity-Induced- Dominancy, see for instance^9,10. As a matter of fact, it is shown that multiple spectral values for time- delay systems can be characterized using a Birkhoff/Vandermonde-based approach; see for instance^11,1,12,13. More precisely, in previous works, it is emphasized that the admissible multiplicity of the real spectral values is bounded by the generic Polya and Szeg¨o bound (denoted P S_B), which is nothing but the degree of the corresponding quasipolynomial (i.e the number of the involved polynomials plus their degree minus one), see for instance¹⁴Problem 206.2, page 144 and page 347. It is worth mentioning that such a bound were recovered

using structured matrices in¹ rather than the principle argument as done in¹⁴. It is important to point out that the multiplicity of a root itself is not important as such but its connection with the dominancy of this root is a meaningful tool for control synthesis. To the best of the authors’ knowledge, the first time an analytical proof of the dominancy of a spectral value for the scalar equation with a single delay was presented and discussed in the 50s, see¹⁵. The dominancy property is further explored and analytically shown in scalar delay equations in¹³, then in second-order systems controlled by a delayed proportional controller is proposed in^16,17 where its applicability in damping active vibrations for a piezo-actuated beam is proved. An extension to the delayed proportional-derivative controller case is studied in^18,19where the dominancy property is parametrically characterized and proven using the argument principle. See also^19,18 which exhibit an analytical proof for the dominancy of the spectral value with maximal multiplicity for second-order systems controlled via a delayed proportional-derivative controller. Recently, in²⁰ it is shown that under appropriate conditions the coexistence of exactlyP SBdistinct negative zeros of quasipolynomial of reduced degree guarantees the exponential stability of the zero solution of the corresponding time-delay system. The dominancy of such real spectral values is shown using an extended factorization technique which generalizes the one provided in²⁰. To the best of the authors’s knowledge the necessary and sufficient conditions derived in the present paper as well as corresponding control strategy represent a novelty.

The present work investigates the effect of structural properties of a class of functional Vandermonde matrices and its effect on qualitative properties of a correspondinglinear autonomous time-delay system of retarded type.

More precisely, the aim of this work is two-fold: firstly, it emphasizes the link between the invertibility of a class of structured functional Vandermonde matrices and the coexistence of distinct real spectral values of linear time-delay systems, which allows to recover the maximal number of distinct real spectral values that may coexist for a given time-delay system. Furthermore, if the number of coexistent real spectral values reaches the P SB, then a necessary and sufficient condition for the asymptotic stability is provided (which is equivalent to the exponential stability²¹p79), see also²² for an estimate of the exponential decay rate for stable linear delay systems. Notice also that the constructive approach we propose, which consists in providing an appropriate factorization of a given quasipolynomial function and then to focus on the location of zeros of one of its factors, gives further insights on such a qualitative property. Namely, it furnishes the exact exponential decay rate rather than just counting the number of the quasipolynomial roots on the left-half plane as may be done by using the principle argument, see for instance⁵.

The class of dynamical systems we consider is ann−th order linear autonomous ordinary differential equa- tions with a forcing term consisting in the delayed dependent variable. This class of systems has an applicative interest particularly in control design problems. As a matter of fact, the forcing term may be seen as a delayed- input able to stabilize the system’s solutions. The idea of exploiting the delay effect in controllers design was first introduced in²³ where it is shown that the conventional proportional controller equipped with an appropriate time-delay performs an averaged derivative action and thus can replace the proportional-derivative controller, see also²⁴. Furthermore, it was stressed in²⁵ that time-delay has a stabilizing effect in the control design.

Indeed, the closed-loop stability is guaranteed precisely by the existence of the delay. Also in²⁶it is shown that a chain ofnintegrators can be stabilized usingndistinct delay blocks, where a delay block is described by two parameters: a ”gain” and a”delay”. The interest of considering control laws of the formPm

k=1γky(t−τk) lies in the simplicity of the controller as well as in its easy practical implementation.

From a control theory point of view, the problem we consider and the approach we propose give rise to anexponential decay assignment method using two parameters a ”gain” and a”delay”. Notice that the idea of using roots assignment for controller-design for time-delay system is not new. For instance, in²⁷ a feedback law yields a finite spectrum of the closed-loop system, located at an arbitrarily preassigned set of points in the complex plane. In the case of systems with delays in control only, a necessary and sufficient condition for finite spectrum assignment is obtained. Notice that the resulting feedback law involves integrals over the past control. In case of delays in state variables it is shown that a technique based on the finite Laplace transform leads to a constructive design procedure. The resulting feedback consists of proportional and (finite interval) integral terms over present and past values of state variables. In²⁸, a similar finite pole placement for time-delay systems with commensurate delays is proposed. Feedback laws defined in terms of Volterra equations are obtained thanks to the properties of the Bezout ring of operators including derivatives, localized and distributed delays. Other analytical/numerical placement methods for retarded time-delay systems are proposed in^29,30, see also³¹ for further insights on pole-placement methods for retarded time-delays systems with proportional-integral-derivative controller-design.

The remaining paper is organized as follows. In Section 2, the problem formulation is presented and some technical lemmas are derived. Section 3 is devoted to the main results of the paper. Section 4 gives an illustrative example showing the applicative perspectives of the derived results. Some concluding remarks end the paper.

Finally, the reader finds proofs of the technical lemmas in the Appendix.

2. Problem settings and prerequisites

In this paper, we are interested in studying the stabilizing effect of the coexistence of the maximal number of real spectral values for the genericn-order ordinary differential equation perturbed by a forcing term depending in the delayed dependent variable

y⁽ⁿ⁾(t) +

n−1

X

k=0

aky^(k)(t) +αy(t−τ) = 0, t∈R⁺, (1) under appropriate initial conditions belonging to the Banach space of continuous functionsC([−τ,0],R) which is an infinite-dimensional differential equation with a single constant delayτ >0.

From a control theory point of view, the aim is to establish a delayed-output-feedback controller u(t) =

−α y(t−τ) able to stabilize solutions of the following control system:

y⁽ⁿ⁾(t) +

n−1

X

k=0

aky^(k)(t) =u(t). (2)

The particular cases of first and second order equations are considered in²⁰, where a stabilizing effect of the coexistence of respectively 2 and 3 negative real roots is shown. By this paper, one generalizes such a result for arbitrary ordern.

In the Laplace domain, the corresponding quasipolynomial characteristic function defined by ∆n:C×R^∗+−→

Cwrites

∆_n(s, τ) :=sⁿ+

n−1

X

k=0

a_ks^k+αe^{−τ s}. (3)

One can prove that the quasipolynomial function (3) admits an infinite number of zeros, see for instance the references^2,32,33. The study of zeros of an entire function³³ of the form (3) plays a crucial role in the analysis of asymptotic stability of the zero solution of Equation (1). Indeed, the zero solution is asymptotically stable if, and only if, all the zeros of (3) are in the open left-half complex plane⁷.

2.1. Counting quasipolynomial roots in horizontal strips

The following result was first introduced and claimed in the problems collection published in 1925 by G.

P´olya and G. Szeg¨o. In the fourth edition of their book¹⁴ Problem 206.2, page 144 and page 347, G. P´olya and G. Szeg¨o emphasize that the proof was obtained by N. Obreschkoff in 1928 using the principle argument, see³⁴. Such a result gives a bound for the number of quasipolynomial’s roots in any horizontal strip. As a consequence, a bound for the number of quasipolynomial’s real roots can be easily deduced.

Theorem 1 (¹⁴). Let τ1, . . . , τN denote real numbers such that τ1 < τ2 < . . . < τN and d1, . . . , dN positive integers such thatd₁+d₂+. . .+d_N =D. Letf_i,j(s)stand for the functionf_i,j(s) =sⁱ⁻¹e^(τjs), for 1≤i≤d_j and1≤j≤N. Let]be the number of zeros of the function

f(s) = X

1≤j≤N 1≤i≤dj

ci,jfi,j(s) (4)

that are contained in the horizontal stripα≤Im(z)≤β. Assuming that X

1≤k≤d1

|ck,1|>0 and X

1≤k≤dN

|ck,N|>0 then

(τN −τ1)(β−α)

2π −D+ 1≤]≤ (τN −τ1)(β−α)

2π +D+N−1. (5)

Settingα=β = 0, the above theorem yields ]_{P S}≤D+N−1 whereD stands for the sum of the degrees of the polynomials involved in the quasipolynomial functionf andN designates the associated number of polynomials.

This gives a sharp bound for the number off’s real roots. Notice thatD+N−1 corresponds to the degree of the quasipolynomialf.¹

1The quasipolynomial degree is defined as the sum of degrees of the involved polynomials plus the corresponding number of delays

Let’s investigate the coexistence ofn+ 1 real (negative) roots for the quasipolynomial ∆_n(., τ). Due to the linearity of ∆_n with respect to its coefficients (a_k)_{0≤k≤n−1} and α, one reduces the system ∆_n(s₁, τ) = . . . =

∆n(sn+1, τ) = 0 to the linear systemVn(Xⁿ⁺¹, τ).V =b where V = (a_n−1, . . . , a0, α)^T, b=−(sⁿ₁, . . . , sⁿ_n+1)^T andX^{n+1 ∆}= (s1, s2,· · ·, sn+1):

Vn Xⁿ⁺¹, τ

=























sⁿ⁻¹₁ sⁿ⁻²₁ · · · s1 1 e^{−τ s1} sⁿ⁻¹₂ sⁿ⁻²₂ · · · s2 1 e^{−τ s2}

... ... . .. ... ... ... sⁿ⁻¹_n sⁿ⁻²_n · · · s_n 1 e^{−τ sn} sⁿ⁻¹_n+1 sⁿ⁻²_n+1 · · · s_n+1 1 e^{−τ sn+1}























. (6)

In the sequel, such a matrix is calledstructured functional Vandermonde type matrix due to its form and its structural properties.

2.2. Structured matrices appearing in the control of dynamical systems

Initially, Birkhoff and Vandermonde matrices are derived from the problem of polynomial interpolation of some unknown functiong, this can be presented in a general way by describing the interpolation conditions in terms ofincidence matrices, see for instance³⁵. For given integersn≥1 andr≥0, the matrix

E =











e1,0 . . . e1,r

... ... en,0 . . . en,r









 ,

is called an incidence matrix ifei,j ∈ {0,1} for everyi and j. Such a matrix contains the data providing the known information about a sufficiently smooth function g : R 7→ R. Let x = (x1, . . . , xn) ∈ Rⁿ such that x1 < . . . < xn, the problem of determining a polynomial ˆP ∈ R[x] with degree less or equal to ι (ι+ 1 = P

1≤i≤n,1≤j≤rei,j) that interpolatesgat (x,E), i.e. which satisfies the conditions:

Pˆ^(j)(x_i) =g^(j)(x_i),

is known as theBirkhoff interpolation problem. Recall thate_i,j= 1 wheng^(j)(x_i) is known, otherwisee_i,j= 0.

Furthermore, an incidence matrix E is said to be poised if such a polynomial ˆP is unique. This amounts to saying that, if, n =Pn

i=1

Pr

j=1ei,j then the coefficients of the interpolating polynomial ˆP are solutions of a linear square system with associated square matrix Υ that we callBirkhoff matrix in the sequel. This matrix is parametrized inxand is shaped byE. It turns out that the incidence matrixE is poised if, and only if, the Birkhoff matrix Υ is non singular for allxsuch thatx1 < . . . < xn. The characterization of poised incidence matrices is solved for interpolation problems of low degrees. As a matter of fact, the problem is still unsolved for any degree greater than six, see for instance^36,37.

Remark 2.1. Unlike Hermite interpolation problem, for which the knowledge of the value of a given order derivative of the interpolating polynomial at a given interpolating point impose the values of all the lower or- ders derivatives of the interpolating polynomial at that point, the Birkhoff interpolation problem release such a restriction. Thereby justifying the qualification of ”lacunary” to describe the Birkhoff interpolation problem.

In the spirit of the definition of functional confluent Vandermonde matrices introduced in³⁸, the following functional Birkhoff matrices were introduced in¹

Definition 2.1. The square functional Birkhoff matrix Υis associated to a sufficiently regular function$ and an incidence matrixE (or equivalently an incidence vectorV) and is defined by:

Υ = [Υ¹Υ² . . .Υ^M]∈ Mδ(R), (7)

where

Υⁱ= [κ^(ki1)(xi)κ^(ki2)(xi). . . κ^(kidi)(xi)], (8) such thatki_l≥0 for all(i, l)∈ {1, . . . , M} × {1, . . . , di}andPM

i=1di=δ where

κ(x_i) =$(x_i)[1. . . x^δ−1_i ]^T, for 1≤i≤M. (9) Analogously to the Birkhoff interpolation problem, in¹the non degeneracy of such functional Birkhoff matrices represent a fondamental assumption for investigating the codimension of the zero spectral values for time-delay systems.

To the best of the author’s knowledge, the first time the Vandermonde matrix appears in a control problem is reported in³⁹ p. 121, where the controllability of a finite dimensional dynamical system is guaranteed by the invertibility of such a matrix, see also^38,40. Next, in the context of time-delay systems, the use of the standard Vandermonde matrix properties was proposed by^26,7 when controlling one chain of integrators by delay blocks.

Here we further exploit the algebraic properties of such structured matrices into a different context.

2.3. The determinant of a structured functional Vandermonde type matrix

The following auxiliary result gives explicitly the determinant of the structured functional Vandermonde type matrix (6). Its proof is presented in the Appendix. In the following we adopt the notation [x, y]_t to designate thet−convex combination of the real (or complex) numbersxandy, that is: [x, y]_t=tx+ (1−t)y fort∈[0,1].

Theorem 2. For any distinct real numbers s_n+1 < · · · < s₂ < s₁, and τ > 0, the structured functional Vandermonde type matrixV_n Xⁿ⁺¹, τ

is invertible. Moreover, its determinant is Qn(Xⁿ⁺¹, τ) = det Vn Xⁿ⁺¹, τ

=τⁿ

n+1

Y

i<j i,j=1

(si−sj)Fτ,n Xⁿ⁺¹

, (10)

which is always positive and whereF_τ,n:Rⁿ⁺¹ →R^∗+ is defined as follows:

F_τ,n Xⁿ⁺¹

=

1

Z

0

· · ·

1

Z

0

| {z }

ntimes n−1

Y

k=1

(1−t_k)^n−k.e

−τ

s₁,[^s2,···[s_n,s_n+1]

tn···]_t

2

t1dt_n· · ·dt₁

Remark 2.2. It is worth mentioning that the product in the expression ofQ_n given by (10)corresponds to the determinant of the standard Vandermonde matrix, see for instance⁴¹.

2.4. Symmetry property

The multivariate functionF_τ,nadmits an invariance property that will be emphasized in the following Lemma which will be used in the proof of the main results. Its proof is presented in the appendix.

Lemma 2.1. For any positive delayτthe functionalFτ,nis invariant for any permutation of the finite sequence (s₁, s₂,· · · , s_n+1),namely, for any permutation σof Xⁿ⁺¹, we have

Fτ,n Xⁿ⁺¹

=Fτ,n σ Xⁿ⁺¹ . For instance, forn= 2, Lemma 2.1 allows to say that for all (x, y, z)∈R³,

Fτ,2(x, y, z) =

1

Z

0 1

Z

0

(1−t1)e^{−τ(t1x+(1−t1) (t2y+(1−t2)z))}dt1dt2

and

F_τ,2(x, y, z) =F_τ,2(x, z, y) =F_τ,2(y, x, z) =F_τ,2(y, z, x) =F_τ,2(z, x, y) =F_τ,2(z, y, x).

Remark 2.3. The symmetry property emphasized in the above Lemma 2.1 is justified by the convexity property on the argument of the exponential kernel. Its proof which can be found in the appendix relies on simple change of coordinates.

2.5. Shifting properties

The following Lemmas exhibit some shifting properties which will be used in the proof of the main results.

Their proofs are presented in the appendix.

Lemma 2.2. Let (si)ⁿ⁺¹_i=1 be a sequence of distinct real numbers. Let(si_k)^k=m+1_k=1 ⊂Rbe any subsequence from (si)ⁿ⁺¹_i=1 ⊂R. For 1≤M ≤n−1,let

Im,M =







(i1, i2,· · ·, im)∈N^m,

m

X

j=1

ij=M





 . Then

X

(i₁,i₂,···,im)∈Im,M

m

Y

k=1

sⁱ_j^k

k− X

(i₁,i₂,···,i_m)∈Im,M

m

Y

k=1

sⁱ_j^k

k+1= s_j1−s_jm+1 X

(i₁,i₂,···,,i_m+1)∈Im+1,M−1

m+1

Y

k=1

sⁱ_j^k

k

Lemma 2.3. Let τ > 0 andn ≥1. Let (s_i)ⁿ⁺¹_i=1 be a sequence of distinct real numbers. For any subsequence (s_ik)^k=m+1_k=1 from (s_i)ⁿ⁺¹_i=1 , the functionF_τ,m satisfies

F_τ,m−1(si₁, si₂,· · ·, si_m)−F_τ,m−1 si₂,· · · , si_m, si_m+1

=−τ si₁−si_m+1

Fτ,m si₁, si₂,· · ·, si_m, si_m+1 . Remark 2.4.

• Lemma 2.2 and Lemma 2.3 remain valid even if the elements of the sequence(si)_1≤i≤n+1 are distinct and complex.

• Under the conditions of Lemma 2.3 it is obvious thatF_τ,n>0 for any τ >0.

2.6. Factorization property

The following Lemma provides a way to factorize a given quasipolynomial function (3) having at least n distinct real roots. This will be used in the proof of the main results.

Lemma 2.4. Assume that the quasipolynomial (3) admits n distinct real roots sn < . . . < s1 then it can be written under the following factorized form:

∆n(s, τ) =

n

Y

i=1

(s−si) [1 + (−τ)ⁿαFτ,n(s, s1,· · ·, sn)]. (11)

3. Main results

In this section, we provide mainly two theorems exploiting the structural properties of the considered class of functional Vandermonde matrices to give some qualitative properties of the solutions of (1). Namely, the first theorem gives conditions on the coexistence of real roots of the quasipolynomial ∆n. The next theorem emphasizes the effect of the coexistence of such real roots on the remaining roots of ∆n. Finally, the combination of those results allows to give some important insights on the exponential stability of the solutions of (1).

3.1. Coexistence ofn+ 1 real roots of∆_n

The following Theorem 3 allows to recoverP SBas a bound of the admissible number of coexisting real roots for the quasipolynomial (3), see for instance¹⁴. This provides an alternative constructive analytical proof based on factorization technique. Furthermore, explicit conditions on the parameters guaranteeing the coexistence of such a number of real roots is provided allowing to Vieta’s-like formulas for quasipolynomials.

Theorem 3.

i) The maximal number of coexisting real roots of the quasipolynomial (3) isn+ 1.

ii) For a fixed τ > 0, Equation (3) admits n+ 1 distinct real spectral values s_n+1, s_n, · · ·, s₂ and s₁ with s_n+1 < · · · < s₂ < s₁ if, and only if, the coefficients (a_k)_{0≤k≤n−1} and α are respectively given by the following functions inτ andXⁿ⁺¹= (s1, . . . , sn+1)

a0 Xⁿ⁺¹, τ

= 1

Qn(Xⁿ⁺¹, τ)det























sⁿ⁻¹₁ sⁿ⁻²₁ · · · s1 −sⁿ₁ e^{−τ s1} sⁿ⁻¹₂ sⁿ⁻²₂ · · · s2 −sⁿ₂ e^{−τ s2}

... ... . .. ... ... ... sⁿ⁻¹_n sⁿ⁻²_n · · · s_n −sⁿ_n e^{−τ sn} sⁿ⁻¹_n+1 sⁿ⁻²_n+1 · · · sn+1 −sⁿ_n+1 e^{−τ sn+1}























, (12)

and for1≤k≤n−1 one has:

ak Xⁿ⁺¹, τ

= 1

Qn(Xⁿ⁺¹, τ)det























sⁿ⁻¹₁ sⁿ⁻²₁ · · · s^k+1₁ −sⁿ₁ s^k−1₁ · · · s1 1 e^{−τ s1} sⁿ⁻¹₂ sⁿ⁻²₂ · · · s^k+1₂ −sⁿ₂ s^k−1₂ · · · s2 1 e^{−τ s2}

... ... . .. ... ... ... . .. ... ... ... sⁿ⁻¹_n sⁿ⁻²_n · · · s^k+1_n −sⁿ_n s^k−1_n · · · sn 1 e^{−τ sn} sⁿ⁻¹_n+1 sⁿ⁻²_n+1 · · · s^k+1_n+1 −sⁿ_n+1 s^k+1_n+1 · · · sn+1 1 e^{−τ sn+1}





















 ,

(13) and

α Xⁿ⁺¹, τ

= 1

Q_n(Xⁿ⁺¹, τ)det























sⁿ⁻¹₁ sⁿ⁻²₁ · · · s1 1 −sⁿ₁ sⁿ⁻¹₂ sⁿ⁻²₂ · · · s2 1 −sⁿ₂ ... ... . .. ... ... ... sⁿ⁻¹_n sⁿ⁻²_n · · · s_n 1 −sⁿ_n sⁿ⁻¹_n+1 sⁿ⁻²_n+1 · · · s_n+1 1 −sⁿ_n+1























. (14)

Remark 3.1.

• From a control theory point of view, let recall the design problem presented in(2), which consists in tuning the controller gainαand the delay parameterτsuch that the closed-loop system’s solution is asymptotically stable. In such a problem the sign of the controller gain is important with respect to the system structure.

Here one has to emphasize that in the design induced from the result we propose, the coefficient α is of alternate sign with respect to the parity of the derivative ordern.

• One can observe that the asymptotic expansion of the coefficientsa_kallows to recover the well-know Vieta’s formulas. This comes from the fact that when τ→ ∞the quasipolynomial ∆_n reduces to a polynomial of degree n. So here the important fact to emphasize is the disappearance of the (n+ 1)-th real root of the quasipolynomial∆_n.

Proof of Theorem 3. Let us start by the proof of ii) and we conclude byi).

ii) Assume that (3) admits n+ 1 real spectral valuess1> s2>· · ·> sn+1.This means that the coefficients

(a_k)_{0≤k≤n−1} andαsatisfy the linear system

∆_n(s_i, τ) =sⁿ_i +

n−1

X

k=0

a_ks^k_i +αe^{−τ si}= 0, for alli= 1,· · ·, n+ 1. (15) Thanks to the invertibility of structured functional Vandermonde type matrix Vn Xⁿ⁺¹, τ

as asserted in Theorem 2, one deals with a Cramer system with respect to the coefficients (ak)_{0≤k≤n−1} and α. So that, one easily computes these coefficients with the standard formulas allowing to (12), (13) and (14) respectively. In particular, the expression ofα Xⁿ⁺¹, τ

is reduced to

α Xⁿ⁺¹, τ

=

(−1)ⁿ⁺¹

n+1

Q

i<j i,j=1

(s_i−s_j)

detV_n(Xⁿ⁺¹, τ) = (−1)ⁿ⁺¹

τⁿF_τ,n Xⁿ⁺¹⁻¹

. (16)

showing the alternating sign ofα.

i) Let proceed by contradiction in assuming the coexistence of n+ 2 real roots of (3). We shall use the factorization of (3) derived in Lemma 2.4, that is:

∆n(s, τ) =

n

Y

i=1

(s−si) [1 + (−τ)ⁿαFτ,n(s, s1,· · ·, sn)]. Since we assumed thats_n+1 ands_n+2are two distinct real roots of ∆_n then one has















∆_n(s_n+1, τ) =

n

Y

i=1

(s_n+1−s_i) [1 + (−τ)ⁿαF_τ,n(s_n+1, s₁,· · · , s_n)] = 0,

∆n(sn+2, τ) =

n

Y

i=1

(sn+2−si) [1 + (−τ)ⁿαFτ,n(sn+2, s1,· · · , sn)] = 0.

(17)

Hence,Fτ,n(sn+2, s1,· · ·, sn)−Fτ,n(sn+2, s1,· · · , sn) = 0, which, using the shift property given in Lemma 2.3, proves the inconsistency in assuming the coexistence ofn+ 2 distinct real roots.

3.2. On qualitative properties ofs1 as a root of∆n

To study the stability of solutions of Equation (3), one needs to study the negativity as well as the dominancy of the roots1 by using an adequate factorization of the quasipolynomial ∆n(s, τ) in (3).

Theorem 4. The following assertions hold:

i) (Negativity)The spectral values1 is negative if, and only if, there existsτ^∗>0 such that an−1(τ^∗) +

n

X

k=2

sk = 0. (18)

ii) (Dominancy)The spectral value s1 is the spectral abscissa of Equation (1).

Proof of Theorem 4.

i) Let assume that s₁ <0. Since the parameter a_n−1 given by (13) is a continuous function with respect to the delay τ and thanks to the l’Hospital’s rule one asserts that its asymptotic behavior is described by: lim_τ→0a_n−1(τ) = −∞and lim_τ→∞a_n−1(τ) =−Pn

k=1sk >0, which proves the existence of τ^∗ >0 such that a_n−1(τ^∗) +Pn

k=2sk = 0. Conversely, to show the negativity of s1, one exploites determinant expressions provided in Theorem 2, allowing to write for anyτ >0 one has:

a_n−1(τ) =−

n

X

k=1

s_k−1 τ

F_n−1,τ(s₁, . . . , s_n) Fn,τ(s1, . . . , sn+1). In particular

a_n−1(τ^∗) +

n

X

k=2

s_k =−s1− 1 τ^∗

F_n−1,τ^∗(s₁, . . . , s_n) Fn,τ^∗(s1, . . . , sn+1).

Using (18) and the positivity ofτ^∗ as well as the positivity of bothFn,τ^∗ andF_n−1,τ^∗ one concludes s₁=−1

τ^∗

F_n−1,τ^∗(s₁, . . . , s_n) Fn,τ^∗(s1, . . . , sn+1) <0.

ii) The proof is based on the quasipolynomial factorization established in the proof of Theorem 3, more precisely, see formula (11).

To prove dominancy property fors1, let us assume that there exists somes0=ζ+jηa root of ∆n(s, τ) = 0 such thatζ > s1. This means thatP(s0, τ) = 0.Hence

1 = (−1)ⁿ⁺¹τⁿαF_τ,n(s₀, s₁,· · ·, s_n) = (−1)ⁿ⁺¹τⁿαRe (F_τ,n(s₀, s₁,· · ·, s_n))

=τⁿ|α|Re (Fτ,n(s0, s1,· · ·, sn))≤τⁿ|α|Fτ,n(ζ, s1,· · · , sn).

(19)

Denote byx2,nthe quantity

s2,· · ·[s_n−1, sn]_t

n· · ·

t3.Rewriting the termh

ζ,[x2,n, s1]_t

2

i

t1

as follows h

ζ,[x2,n, s1]_t

2

i

t1

= t1(ζ−s1) +s1+t2(1−t1) (x2,n−s1)

= t1(ζ−s1) + [x2,n, s1]_t

2(1−t1)

= t1(ζ−s1) +h

s1,[x2,n, s1]_t

2

i

t₁. Then, using the following estimates

h

s1,[x2,n, s1]_t

2

i

t₁>h

s1,[x2,n, sn+1]_t

2

i

t₁ and e^{−τ t1(ζ−s1)}<1, ∀t1∈]0,1[

we get from (19) and Lemma 2.1 1 ≤ τⁿ|α|

1

Z

0

· · ·

1

Z

0

| {z }

n times n−1

Y

k=1

(1−tk)^n−ke^{−τ t1(ζ−s1)}e^−τ

hζ,[x_2,n,s₁]_t

2

i

t1dtn· · ·dt1

< τⁿ|α|

1

Z

0

· · ·

1

Z

0

| {z }

n times n−1

Y

k=1

(1−t_k)^n−ke^−τ

h

s₁,[x_2,n,s_n+1]_t

2

i

t1dt_n· · ·dt₁

= τⁿ|α|F_τ,n(s₁, s₂,· · ·, s_n+1) = 1 (thanks to (16)),

which is inconsistent. This proves the dominancy ofs1. The proof of Theorem 4 is achieved.

Remark 3.2. Note that the factorization (11)of∆n(., τ)allows to retrieve the explicit expression of the coef- ficientαdefined in (14), since sn+1 is a root of quasipolynomial ∆n(., τ).Just replace sbysn+1 in(11). 3.3. Exponential stability

Note that for a linear retarded functional differential equations the exponential stability is equivalent to the uniform asymptotic stability,²¹ p79. Further, for the linear autonomous retarded functional differential equations, asymptotic stability implies uniform asymptotic stability and, hence, exponential stability. Recall that Theorem 3 gives necessary and sufficient conditions for the coexistence ofn+ 1 real roots of (3). Theorem 4 gives a necessary and sufficient conditions for the negativity of all such real roots and asserts that the roots of (3) have necessarily<(s)< s1. So the following result which is a direct consequence of Theorems 3-4 allows to the exponential stability.

Corollary 3.1. If equation (3)admits(n+ 1)distinct real spectral valuessn+1< . . . < s1 and (18)is satisfied then the trivial solution of (1)is exponentially stable with s₁ as a decay rate.

4. Stabilizing coupled oscillators using delayed output feedback

To show the applicative potential of the obtained results, let consider as an illustrative example a system consisting in two coupled oscillators. Coupled oscillations occur when two or more oscillating systems are connected in such a way the motion energy is transferred between them. The dynamics of coupled oscillators plays an important role in a variety of systems in nature and technology, see for instance⁴² and references therein. Their ability to display complex self-organized dynamical phenomena makes them an important tool to explain fundamental mechanism of emergent dynamics in coupled systems. It is known that when the coupling is small then each oscillator operates at its natural frequency and the system is then said to be incoherent.

However, when the coupling exceeds a certain threshold then the system spontaneously synchronizes. Here we consider the mechanical system of two coupled oscillators as depicted in Figure 1 and we aim to design a stabilizing delayed controller, which corresponds to oscillation quenching. Using the fundamental principle of dynamics and the standard assumption about the linearity of the damping lead to the following differential equations governing the motion of the system:







m₁x¨₁(t) =−b1x˙₁(t)−k₁x₁(t) +k₂(x₂(t)−x₁(t)) +f(t), m2x¨2(t) =−k2(x2(t)−x1(t)).

(20)

Figure 1: Coupled damped oscillators.

where the parameters values are chosen accordingly to an experimental settings: b1 = ¹₂, k1 = 0.836, k2 = 1, m1 = 0.15, m2 = 3. If forcing term f acts on the system as an input and takes a proportional-minus-delay structure as suggested in²³; that isf(t) =−α0x₂(t)−α1x₂(t−τ) and by settingξ(t) = (x₁(t),x˙₁(t), x₂(t),x˙₂(t)) the above system writes

ξ(t) =˙ A₀ξ(t) +A₁ξ(t−τ), (21)

where

A0=























0 1 0 0

−^k2_m^+k1

1 −_m^b1

1

k₂ m1 −_m^α0

1 0

0 0 0 1

k₂

m₂ 0 −_m^k2

2 0























and A1=























0 0 0 0

0 0 −_m^α1

1 0

0 0 0 0

0 0 0 0





















 .

The corresponding characteristic quasipolynomial function has the form (3) and writes explicitely as follows:

∆4(s, τ) =s⁴+ b₁ m1

s³+(k₂m₂+m₁k₂+k₁m₂) m1m2

s²+ b₁k₂ m1m2

s+k₁k₂+k₂α₀ m1m2

+k₂α₁e^−sτ m1m2

. (22)

The aim is to establish values for controller’s gains α₀ and α₁ as well as the value of the delay parameterτ enabling us to assignP S_B = 5 real roots of the quasipolynomial (22) guaranteeing the exponential stability of the trivial solution of the closed-loop system as asserted in Theorem 4. To simplify the design task, we consider the case of equidistributed negative spectral values where the distance between two consecutive roots is d= ¹₂. By setting a targeted decay rate or equivalently the rightmost root, for instance at s1 =−1; that issk =s1−^k−1₂ fork= 2, . . . ,5 one then applies Theorem 4 and a simple parameter identification to recover the gains valuesα0 ≈5.29, α1≈ −4.54 and the delay valueτ ≈0.81, the spectrum distribution illustrated in Figure 2.

5. Concluding remarks

By this paper, we investigated conditions on the coefficients of then−thorder linear ordinary differential equations with delayed-state forcing term guaranteeing the coexistence of the maximal number of real spectral

Figure 2: Spectrum distribution of the closed-loop system (21) using a proportional-minus-delay controller. The parameters values are given in Section 4.

values, which itself corresponds to the well-known Polya and Szeg¨o bound for quasipolynomial’s real roots. Such a bound was recovered using an analytical constructive approach. Furthermore, an easy to check criterion was provided, which allows to characterize the stabilizing effect of the coexistence of such spectral values. It is worth noting that such a configuration guarantees the exponential stability and explicitly describes the corresponding exponential decay rate. The applicative potential of the presented results is illustrated through the problem of stabilizing controller design for the system of coupled oscillators.

Acknowledgements

IB & SIN are partially financially supported by a public grant overseen by the French National Research Agency (ANR) as part of the ”Investissement d’Avenir” program, through the ”iCODE Institute project” funded by the IDEX Paris-Saclay and a grant from Hubert Curien (PHC) BALATON, project number 40502NM.

The authors warmly thank Jean-Jacques Loiseau (LS2N Nantes, France) for discussion and insights on pole- placement for delay systems.

Appendix: Proof of the technical lemmas

Proof of Lemma 2.1. Without loss of generality, it suffices to consider the following permutation σ_i(s₁, s₂,· · · , s_i, s_i+1,· · ·, s_n+1) = (s₁, s₂,· · · , s_i+1, s_i,· · ·, s_n+1),

where 1 ≤ i ≤ n. Any other permutation of sets of indices is none other than the composition of such permutations. For example, ifσ_i,j, withj−i >1, is such that

σi,j(s1, s2,· · · , si,· · · , sj,· · · , sn+1) = (s1, s2,· · ·, sj,· · ·, si,· · · , sn+1), then

σi,j=σi◦σi+1◦ · · · ◦σ_j−2◦σ_j−1◦ · · · ◦σi+1◦σi. Writeh

s1,

s2,· · ·[sn, sn+1]_t

n· · ·

t₂

i

t₁ as t1s1+ (1−t1)t2s2+· · ·+

i−1

Y

k=1

(1−tk)tisi+

i

Y

k=1

(1−tk)ti+1si+1+· · ·+

n−1

Y

k=1

(1−tk)tnsn+

n

Y

k=1

(1−tk)sn+1. It is then necessary to introduce a suitable change of variable, that switches the coefficient of s_i with the coefficient ofs_i+1, without affecting the other coefficients. Let





















































uk = tk, k6=i∧i+ 1, ui = (1−ti)ti+1

u_i+1 = t_i

1−ti+1+titi+1

if 1≤i≤n−1

and







uk = tk, 1≤k≤n−1 u_n = 1−t_n

if i=n

(23)

Clearly,u_i∈]0,1[ for all 1≤i≤n−1.Moreover, from (1−t_i) (1−t_i+1)>0,we have 1−t_i+1+t_it_i+1 >

ti>0, henceui+1∈]0,1[. The jacobian matrixJ = ^D(u_D(t^{1,u2,···,un)}

1,t₂,···,t_n) is such that detJ= ti−1

t_it_i+1−t_i+1+ 1 6= 0.

So, (23) defines aC¹−diffeomorphism from ]0,1[ⁿ into ]0,1[ⁿ,for all 1≤i≤n−1,and the following properties ti

i−1

Q

k=1

(1−tk) =ui+1 i

Q

k=1

(1−uk), t_i+1

i

Q

k=1

(1−t_k) =u_i

i−1

Q

k=1

(1−u_k), tm

m−1

Q

k=1

(1−tk) =um m−1

Q

k=1

(1−uk), ∀m∈ {2,· · · , n}, m6=i∧i+ 1 are satisfied.

On the other hand, from du₁du₂· · ·du_n=

detD(u1, u2,· · ·, un) D(t₁, t₂,· · ·, t_n)

dt₁dt₂· · ·dt_n= 1−ti

1−u_idt₁dt₂· · ·dt_n,